# Properties

 Label 1512.2.c.d Level 1512 Weight 2 Character orbit 1512.c Analytic conductor 12.073 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1512 = 2^{3} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1512.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.0733807856$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\Q(\zeta_{40})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}\cdot 3^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{40}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{40} - \zeta_{40}^{5} - \zeta_{40}^{7} + \zeta_{40}^{9} - \zeta_{40}^{13} ) q^{2} -2 \zeta_{40}^{4} q^{4} + ( \zeta_{40} - \zeta_{40}^{3} + \zeta_{40}^{5} - \zeta_{40}^{11} - \zeta_{40}^{13} + 2 \zeta_{40}^{15} ) q^{5} - q^{7} + ( -2 \zeta_{40} + 2 \zeta_{40}^{11} ) q^{8} +O(q^{10})$$ $$q + ( \zeta_{40} - \zeta_{40}^{5} - \zeta_{40}^{7} + \zeta_{40}^{9} - \zeta_{40}^{13} ) q^{2} -2 \zeta_{40}^{4} q^{4} + ( \zeta_{40} - \zeta_{40}^{3} + \zeta_{40}^{5} - \zeta_{40}^{11} - \zeta_{40}^{13} + 2 \zeta_{40}^{15} ) q^{5} - q^{7} + ( -2 \zeta_{40} + 2 \zeta_{40}^{11} ) q^{8} + ( -2 + 3 \zeta_{40}^{2} - 2 \zeta_{40}^{8} + \zeta_{40}^{12} ) q^{10} + ( 2 \zeta_{40}^{5} + \zeta_{40}^{7} - \zeta_{40}^{9} - \zeta_{40}^{11} + \zeta_{40}^{13} + 2 \zeta_{40}^{15} ) q^{11} + ( 1 - 2 \zeta_{40}^{4} + 3 \zeta_{40}^{6} + \zeta_{40}^{8} - \zeta_{40}^{12} + 3 \zeta_{40}^{14} ) q^{13} + ( -\zeta_{40} + \zeta_{40}^{5} + \zeta_{40}^{7} - \zeta_{40}^{9} + \zeta_{40}^{13} ) q^{14} + 4 \zeta_{40}^{8} q^{16} + ( 2 \zeta_{40} + 2 \zeta_{40}^{3} - \zeta_{40}^{5} - 2 \zeta_{40}^{7} + 2 \zeta_{40}^{9} - \zeta_{40}^{15} ) q^{17} + ( 1 - 2 \zeta_{40}^{4} - 2 \zeta_{40}^{12} ) q^{19} + ( -2 \zeta_{40} + 4 \zeta_{40}^{3} - 2 \zeta_{40}^{7} - 4 \zeta_{40}^{9} + 4 \zeta_{40}^{11} + 2 \zeta_{40}^{13} - 2 \zeta_{40}^{15} ) q^{20} + ( 3 \zeta_{40}^{2} + 2 \zeta_{40}^{4} - 2 \zeta_{40}^{8} + \zeta_{40}^{12} ) q^{22} + ( \zeta_{40} + \zeta_{40}^{3} + 3 \zeta_{40}^{5} - 4 \zeta_{40}^{7} - 2 \zeta_{40}^{9} + 3 \zeta_{40}^{11} + 3 \zeta_{40}^{13} - 4 \zeta_{40}^{15} ) q^{23} + ( -3 + 6 \zeta_{40}^{2} - 2 \zeta_{40}^{8} + 3 \zeta_{40}^{10} + 2 \zeta_{40}^{12} - 6 \zeta_{40}^{14} ) q^{25} + ( 2 \zeta_{40} + 2 \zeta_{40}^{3} + 4 \zeta_{40}^{11} - 4 \zeta_{40}^{13} ) q^{26} + 2 \zeta_{40}^{4} q^{28} + ( -\zeta_{40} + \zeta_{40}^{3} - \zeta_{40}^{7} - 5 \zeta_{40}^{9} - 4 \zeta_{40}^{11} - \zeta_{40}^{15} ) q^{29} + ( 3 + 6 \zeta_{40}^{2} - 3 \zeta_{40}^{6} + \zeta_{40}^{8} + 3 \zeta_{40}^{10} - \zeta_{40}^{12} - 3 \zeta_{40}^{14} ) q^{31} + ( 4 \zeta_{40}^{5} - 4 \zeta_{40}^{15} ) q^{32} + ( 4 - 4 \zeta_{40}^{4} - 2 \zeta_{40}^{12} ) q^{34} + ( -\zeta_{40} + \zeta_{40}^{3} - \zeta_{40}^{5} + \zeta_{40}^{11} + \zeta_{40}^{13} - 2 \zeta_{40}^{15} ) q^{35} + ( -1 + 2 \zeta_{40}^{4} + 3 \zeta_{40}^{6} - 3 \zeta_{40}^{8} - 3 \zeta_{40}^{10} - \zeta_{40}^{12} + 3 \zeta_{40}^{14} ) q^{37} + ( -\zeta_{40} - 2 \zeta_{40}^{3} - \zeta_{40}^{5} + \zeta_{40}^{7} - \zeta_{40}^{9} - \zeta_{40}^{13} + 2 \zeta_{40}^{15} ) q^{38} + ( 2 + 2 \zeta_{40}^{4} - 6 \zeta_{40}^{6} + 2 \zeta_{40}^{8} + 2 \zeta_{40}^{12} ) q^{40} + ( 2 \zeta_{40} + 2 \zeta_{40}^{3} + \zeta_{40}^{7} - \zeta_{40}^{9} + 3 \zeta_{40}^{11} - 3 \zeta_{40}^{13} - 2 \zeta_{40}^{15} ) q^{41} + ( -3 + 6 \zeta_{40}^{4} + 3 \zeta_{40}^{6} + \zeta_{40}^{8} - 6 \zeta_{40}^{10} + 7 \zeta_{40}^{12} + 3 \zeta_{40}^{14} ) q^{43} + ( 2 \zeta_{40} + 4 \zeta_{40}^{3} - 2 \zeta_{40}^{5} - 4 \zeta_{40}^{7} - 2 \zeta_{40}^{9} + 2 \zeta_{40}^{11} - 2 \zeta_{40}^{15} ) q^{44} + ( 2 + 3 \zeta_{40}^{2} - 2 \zeta_{40}^{4} - 6 \zeta_{40}^{6} + 6 \zeta_{40}^{10} - 5 \zeta_{40}^{12} ) q^{46} + ( -3 \zeta_{40} - 3 \zeta_{40}^{3} - 3 \zeta_{40}^{7} - 3 \zeta_{40}^{9} + 6 \zeta_{40}^{13} + 3 \zeta_{40}^{15} ) q^{47} + q^{49} + ( -6 \zeta_{40} + 8 \zeta_{40}^{3} - 2 \zeta_{40}^{5} - 2 \zeta_{40}^{7} - 4 \zeta_{40}^{9} + 2 \zeta_{40}^{11} - 6 \zeta_{40}^{15} ) q^{50} + ( -2 + 6 \zeta_{40}^{2} - 6 \zeta_{40}^{6} + 2 \zeta_{40}^{8} - 6 \zeta_{40}^{14} ) q^{52} + ( -6 \zeta_{40} + 6 \zeta_{40}^{3} + 4 \zeta_{40}^{5} - 4 \zeta_{40}^{7} - 2 \zeta_{40}^{9} + 4 \zeta_{40}^{11} + 2 \zeta_{40}^{13} - 2 \zeta_{40}^{15} ) q^{53} + ( -4 + 3 \zeta_{40}^{6} + \zeta_{40}^{8} - \zeta_{40}^{12} - 3 \zeta_{40}^{14} ) q^{55} + ( 2 \zeta_{40} - 2 \zeta_{40}^{11} ) q^{56} + ( -4 - 6 \zeta_{40}^{2} + 4 \zeta_{40}^{4} - 8 \zeta_{40}^{8} - 6 \zeta_{40}^{10} + 4 \zeta_{40}^{12} + 6 \zeta_{40}^{14} ) q^{58} + ( \zeta_{40} - \zeta_{40}^{3} - 4 \zeta_{40}^{5} - \zeta_{40}^{7} + \zeta_{40}^{9} - 2 \zeta_{40}^{13} - 3 \zeta_{40}^{15} ) q^{59} + ( -2 + 4 \zeta_{40}^{4} - 2 \zeta_{40}^{8} + 6 \zeta_{40}^{10} + 2 \zeta_{40}^{12} ) q^{61} + ( 3 \zeta_{40} + 2 \zeta_{40}^{3} - 5 \zeta_{40}^{5} - 5 \zeta_{40}^{7} - \zeta_{40}^{9} + 2 \zeta_{40}^{11} - 3 \zeta_{40}^{13} - 6 \zeta_{40}^{15} ) q^{62} -8 \zeta_{40}^{12} q^{64} + ( \zeta_{40} + \zeta_{40}^{3} - 4 \zeta_{40}^{5} + 5 \zeta_{40}^{7} - 5 \zeta_{40}^{9} + 6 \zeta_{40}^{11} - 6 \zeta_{40}^{13} + 3 \zeta_{40}^{15} ) q^{65} + ( 3 - 6 \zeta_{40}^{4} + 3 \zeta_{40}^{6} + 7 \zeta_{40}^{8} + \zeta_{40}^{12} + 3 \zeta_{40}^{14} ) q^{67} + ( -2 \zeta_{40}^{3} - 4 \zeta_{40}^{5} - 2 \zeta_{40}^{7} + 2 \zeta_{40}^{9} + 2 \zeta_{40}^{11} - 4 \zeta_{40}^{13} + 2 \zeta_{40}^{15} ) q^{68} + ( 2 - 3 \zeta_{40}^{2} + 2 \zeta_{40}^{8} - \zeta_{40}^{12} ) q^{70} + ( \zeta_{40} + \zeta_{40}^{3} + 4 \zeta_{40}^{5} - 4 \zeta_{40}^{7} - 2 \zeta_{40}^{9} + 3 \zeta_{40}^{11} + 3 \zeta_{40}^{13} - 5 \zeta_{40}^{15} ) q^{71} + ( 3 + 3 \zeta_{40}^{6} + 3 \zeta_{40}^{8} - 3 \zeta_{40}^{12} - 3 \zeta_{40}^{14} ) q^{73} + ( \zeta_{40} + 2 \zeta_{40}^{3} + \zeta_{40}^{5} - \zeta_{40}^{7} - 5 \zeta_{40}^{9} + \zeta_{40}^{13} + 4 \zeta_{40}^{15} ) q^{74} + ( -4 + 2 \zeta_{40}^{4} + 4 \zeta_{40}^{12} ) q^{76} + ( -2 \zeta_{40}^{5} - \zeta_{40}^{7} + \zeta_{40}^{9} + \zeta_{40}^{11} - \zeta_{40}^{13} - 2 \zeta_{40}^{15} ) q^{77} + ( 3 - 12 \zeta_{40}^{2} + 9 \zeta_{40}^{6} + 3 \zeta_{40}^{8} - 6 \zeta_{40}^{10} - 3 \zeta_{40}^{12} + 3 \zeta_{40}^{14} ) q^{79} + ( 4 \zeta_{40} - 4 \zeta_{40}^{3} - 4 \zeta_{40}^{7} + 4 \zeta_{40}^{9} + 4 \zeta_{40}^{13} - 4 \zeta_{40}^{15} ) q^{80} + ( -2 + 3 \zeta_{40}^{2} + 2 \zeta_{40}^{4} - 6 \zeta_{40}^{6} - \zeta_{40}^{12} - 6 \zeta_{40}^{14} ) q^{82} + ( 7 \zeta_{40} - 7 \zeta_{40}^{3} - 8 \zeta_{40}^{5} + 3 \zeta_{40}^{7} + 3 \zeta_{40}^{9} - 4 \zeta_{40}^{11} - 4 \zeta_{40}^{13} - \zeta_{40}^{15} ) q^{83} + ( 1 - 2 \zeta_{40}^{4} + 6 \zeta_{40}^{6} - 2 \zeta_{40}^{8} - 3 \zeta_{40}^{10} - 4 \zeta_{40}^{12} + 6 \zeta_{40}^{14} ) q^{85} + ( 10 \zeta_{40}^{3} + 10 \zeta_{40}^{5} - 10 \zeta_{40}^{7} - 2 \zeta_{40}^{9} + 4 \zeta_{40}^{11} + 6 \zeta_{40}^{13} - 8 \zeta_{40}^{15} ) q^{86} + ( 2 - 2 \zeta_{40}^{4} - 6 \zeta_{40}^{6} - 2 \zeta_{40}^{8} + 2 \zeta_{40}^{12} ) q^{88} + ( 4 \zeta_{40} + 4 \zeta_{40}^{3} - \zeta_{40}^{7} + 7 \zeta_{40}^{9} - 3 \zeta_{40}^{11} - 3 \zeta_{40}^{13} - 4 \zeta_{40}^{15} ) q^{89} + ( -1 + 2 \zeta_{40}^{4} - 3 \zeta_{40}^{6} - \zeta_{40}^{8} + \zeta_{40}^{12} - 3 \zeta_{40}^{14} ) q^{91} + ( 6 \zeta_{40} - 8 \zeta_{40}^{3} - 8 \zeta_{40}^{5} + 6 \zeta_{40}^{7} - 2 \zeta_{40}^{13} + 2 \zeta_{40}^{15} ) q^{92} + ( 6 \zeta_{40}^{10} + 6 \zeta_{40}^{12} + 6 \zeta_{40}^{14} ) q^{94} + ( \zeta_{40} + \zeta_{40}^{3} - 3 \zeta_{40}^{5} + 2 \zeta_{40}^{7} - 2 \zeta_{40}^{9} + 3 \zeta_{40}^{11} - 3 \zeta_{40}^{13} + 2 \zeta_{40}^{15} ) q^{95} + ( -4 - 10 \zeta_{40}^{8} + 10 \zeta_{40}^{12} ) q^{97} + ( \zeta_{40} - \zeta_{40}^{5} - \zeta_{40}^{7} + \zeta_{40}^{9} - \zeta_{40}^{13} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 8q^{4} - 16q^{7} + O(q^{10})$$ $$16q - 8q^{4} - 16q^{7} - 20q^{10} - 16q^{16} + 20q^{22} - 32q^{25} + 8q^{28} + 40q^{31} + 40q^{34} + 40q^{40} + 4q^{46} + 16q^{49} - 40q^{52} - 72q^{55} - 32q^{64} + 20q^{70} + 24q^{73} - 40q^{76} + 24q^{79} - 28q^{82} + 40q^{88} + 24q^{94} + 16q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$$.

 $$n$$ $$757$$ $$785$$ $$1081$$ $$1135$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
757.1
 0.453990 + 0.891007i −0.891007 + 0.453990i −0.891007 − 0.453990i 0.453990 − 0.891007i −0.156434 − 0.987688i −0.987688 + 0.156434i −0.987688 − 0.156434i −0.156434 + 0.987688i 0.156434 − 0.987688i 0.987688 + 0.156434i 0.987688 − 0.156434i 0.156434 + 0.987688i −0.453990 + 0.891007i 0.891007 + 0.453990i 0.891007 − 0.453990i −0.453990 − 0.891007i
−1.14412 0.831254i 0 0.618034 + 1.90211i 1.60758i 0 −1.00000 0.874032 2.68999i 0 −1.33630 + 1.83927i
757.2 −1.14412 0.831254i 0 0.618034 + 1.90211i 2.63506i 0 −1.00000 0.874032 2.68999i 0 2.19041 3.01484i
757.3 −1.14412 + 0.831254i 0 0.618034 1.90211i 2.63506i 0 −1.00000 0.874032 + 2.68999i 0 2.19041 + 3.01484i
757.4 −1.14412 + 0.831254i 0 0.618034 1.90211i 1.60758i 0 −1.00000 0.874032 + 2.68999i 0 −1.33630 1.83927i
757.5 −0.437016 1.34500i 0 −1.61803 + 1.17557i 4.29757i 0 −1.00000 2.28825 + 1.66251i 0 −5.78022 + 1.87811i
757.6 −0.437016 1.34500i 0 −1.61803 + 1.17557i 0.0549306i 0 −1.00000 2.28825 + 1.66251i 0 −0.0738814 + 0.0240055i
757.7 −0.437016 + 1.34500i 0 −1.61803 1.17557i 0.0549306i 0 −1.00000 2.28825 1.66251i 0 −0.0738814 0.0240055i
757.8 −0.437016 + 1.34500i 0 −1.61803 1.17557i 4.29757i 0 −1.00000 2.28825 1.66251i 0 −5.78022 1.87811i
757.9 0.437016 1.34500i 0 −1.61803 1.17557i 4.29757i 0 −1.00000 −2.28825 + 1.66251i 0 −5.78022 1.87811i
757.10 0.437016 1.34500i 0 −1.61803 1.17557i 0.0549306i 0 −1.00000 −2.28825 + 1.66251i 0 −0.0738814 0.0240055i
757.11 0.437016 + 1.34500i 0 −1.61803 + 1.17557i 0.0549306i 0 −1.00000 −2.28825 1.66251i 0 −0.0738814 + 0.0240055i
757.12 0.437016 + 1.34500i 0 −1.61803 + 1.17557i 4.29757i 0 −1.00000 −2.28825 1.66251i 0 −5.78022 + 1.87811i
757.13 1.14412 0.831254i 0 0.618034 1.90211i 1.60758i 0 −1.00000 −0.874032 2.68999i 0 −1.33630 1.83927i
757.14 1.14412 0.831254i 0 0.618034 1.90211i 2.63506i 0 −1.00000 −0.874032 2.68999i 0 2.19041 + 3.01484i
757.15 1.14412 + 0.831254i 0 0.618034 + 1.90211i 2.63506i 0 −1.00000 −0.874032 + 2.68999i 0 2.19041 3.01484i
757.16 1.14412 + 0.831254i 0 0.618034 + 1.90211i 1.60758i 0 −1.00000 −0.874032 + 2.68999i 0 −1.33630 + 1.83927i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 757.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.c.d 16
3.b odd 2 1 inner 1512.2.c.d 16
4.b odd 2 1 6048.2.c.d 16
8.b even 2 1 inner 1512.2.c.d 16
8.d odd 2 1 6048.2.c.d 16
12.b even 2 1 6048.2.c.d 16
24.f even 2 1 6048.2.c.d 16
24.h odd 2 1 inner 1512.2.c.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.c.d 16 1.a even 1 1 trivial
1512.2.c.d 16 3.b odd 2 1 inner
1512.2.c.d 16 8.b even 2 1 inner
1512.2.c.d 16 24.h odd 2 1 inner
6048.2.c.d 16 4.b odd 2 1
6048.2.c.d 16 8.d odd 2 1
6048.2.c.d 16 12.b even 2 1
6048.2.c.d 16 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1512, [\chi])$$:

 $$T_{5}^{8} + 28 T_{5}^{6} + 194 T_{5}^{4} + 332 T_{5}^{2} + 1$$ $$T_{17}^{2} - 10$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} + 4 T^{4} + 8 T^{6} + 16 T^{8} )^{2}$$
$3$ 
$5$ $$( 1 - 12 T^{2} + 54 T^{4} - 48 T^{6} - 469 T^{8} - 1200 T^{10} + 33750 T^{12} - 187500 T^{14} + 390625 T^{16} )^{2}$$
$7$ $$( 1 + T )^{16}$$
$11$ $$( 1 - 60 T^{2} + 1734 T^{4} - 31920 T^{6} + 413051 T^{8} - 3862320 T^{10} + 25387494 T^{12} - 106293660 T^{14} + 214358881 T^{16} )^{2}$$
$13$ $$( 1 - 40 T^{2} + 936 T^{4} - 16040 T^{6} + 219326 T^{8} - 2710760 T^{10} + 26733096 T^{12} - 193072360 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 + 24 T^{2} + 289 T^{4} )^{8}$$
$19$ $$( 1 - 66 T^{2} + 1791 T^{4} - 23826 T^{6} + 130321 T^{8} )^{4}$$
$23$ $$( 1 + 52 T^{2} + 2430 T^{4} + 71312 T^{6} + 1906499 T^{8} + 37724048 T^{10} + 680013630 T^{12} + 7697866228 T^{14} + 78310985281 T^{16} )^{2}$$
$29$ $$( 1 - 44 T^{2} + 960 T^{4} - 18676 T^{6} + 833774 T^{8} - 15706516 T^{10} + 678989760 T^{12} - 26172226124 T^{14} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 - 10 T + 114 T^{2} - 800 T^{3} + 5351 T^{4} - 24800 T^{5} + 109554 T^{6} - 297910 T^{7} + 923521 T^{8} )^{4}$$
$37$ $$( 1 - 192 T^{2} + 17010 T^{4} - 960672 T^{6} + 40219139 T^{8} - 1315159968 T^{10} + 31879478610 T^{12} - 492619470528 T^{14} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 + 220 T^{2} + 24294 T^{4} + 1713680 T^{6} + 83598011 T^{8} + 2880696080 T^{10} + 68649037734 T^{12} + 1045022933020 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 + 32 T^{2} + 4920 T^{4} + 158992 T^{6} + 11619614 T^{8} + 293976208 T^{10} + 16820500920 T^{12} + 202283617568 T^{14} + 11688200277601 T^{16} )^{2}$$
$47$ $$( 1 + 124 T^{2} + 7312 T^{4} + 372292 T^{6} + 18784750 T^{8} + 822393028 T^{10} + 35680227472 T^{12} + 1336622700796 T^{14} + 23811286661761 T^{16} )^{2}$$
$53$ $$( 1 - 216 T^{2} + 24972 T^{4} - 2015208 T^{6} + 122805830 T^{8} - 5660719272 T^{10} + 197041091532 T^{12} - 4787502003864 T^{14} + 62259690411361 T^{16} )^{2}$$
$59$ $$( 1 - 380 T^{2} + 67344 T^{4} - 7257700 T^{6} + 519439886 T^{8} - 25264053700 T^{10} + 816031559184 T^{12} - 16028602783580 T^{14} + 146830437604321 T^{16} )^{2}$$
$61$ $$( 1 - 304 T^{2} + 46620 T^{4} - 4694096 T^{6} + 336476774 T^{8} - 17466731216 T^{10} + 645493107420 T^{12} - 15662193805744 T^{14} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 - 232 T^{2} + 32520 T^{4} - 3258632 T^{6} + 244468094 T^{8} - 14627999048 T^{10} + 655314454920 T^{12} - 20986344663208 T^{14} + 406067677556641 T^{16} )^{2}$$
$71$ $$( 1 + 396 T^{2} + 77350 T^{4} + 9537264 T^{6} + 808347579 T^{8} + 48077347824 T^{10} + 1965593525350 T^{12} + 50727712432716 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 - 6 T + 238 T^{2} - 990 T^{3} + 23766 T^{4} - 72270 T^{5} + 1268302 T^{6} - 2334102 T^{7} + 28398241 T^{8} )^{4}$$
$79$ $$( 1 - 6 T + 82 T^{2} - 18 T^{3} + 6630 T^{4} - 1422 T^{5} + 511762 T^{6} - 2958234 T^{7} + 38950081 T^{8} )^{4}$$
$83$ $$( 1 - 252 T^{2} + 47280 T^{4} - 5629572 T^{6} + 552317774 T^{8} - 38782121508 T^{10} + 2243829416880 T^{12} - 82388974088988 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 + 364 T^{2} + 76230 T^{4} + 10653296 T^{6} + 1102200539 T^{8} + 84384757616 T^{10} + 4782841031430 T^{12} + 180901189909804 T^{14} + 3936588805702081 T^{16} )^{2}$$
$97$ $$( 1 - 2 T + 70 T^{2} - 194 T^{3} + 9409 T^{4} )^{8}$$